Time-varying exposure to permanent and short-term risk and stock price momentum

Transcription

1 Time-varying exposure to permanent and short-term risk and stock price momentum Elisa Pazaj Abstract This paper provides an explanation for the documented link between earnings and stock price momentum. A simple dynamic model that accounts for firms exposures to both short-term and long-term earnings shocks produces momentum-like effects. Price sensitivities to the two types of shocks vary in the cross-section depending on firm fundamentals and change over time depending on firm financial health. The combined effects of short-term and long-term earnings shocks lead to a positive relationship between cumulative past and expected returns of winner and loser momentum portfolios. The model predicts greater profitability of momentum strategies in the subset of companies that are more sensitive to short-term shocks and temporarily financially constrained. Empirical tests support the model predictions. Keywords: Momentum, short-term and long-term earnings shocks, cash holdings. JEL Classification: G12, G32, G35. Cass Business School. Elisa.Pazaj.1@cass.city.ac.uk

2 I. Introduction Momentum investment strategies go long the stocks with the highest recent price performance and short the ones with the lowest. The profitability of these so called momentum portfolios was first documented by Jegadeesh and Titman (1993) for the US stock market. A large body of literature has since shown the presence of momentum in other markets around the world. 1 A theory of the momentum premium should be able to not only justify the magnitude of the strategy s profits but also the reversal in returns for holding periods beyond a year. Both sentiment and rational theories have been proposed in the literature. 2 Explanations based on sentiment rely on biases in the way investors process information. Barberis, Schleifer and Vishny (1998) argue that a conservatism bias, which implies that investors are slow in updating their beliefs in the face of new information, leads to under-reaction in the short term. Daniel, Hirschleifer and Subrahmanyam (1998) present a model where overconfident traders overweight past information deemed to be correct, neglecting future information. Both models can generate price continuations and reversals but they cannot explain the magnitude and the specific formation and holding periods over which momentum strategies are profitable. Rational explanations argue that returns to momentum strategies are a compensation for risk. This can be the case if there is a dispersion in unconditional drifts in returns (Conrad and Kaul, 1998), or if betas are time-varying but persistent. 3 While rational models can provide quantitative predictions regarding the magnitude and performance patterns of momentum strategies, they are not successful in replicating both simultaneously. A relatively more recent empirical literature (Chordia and Shivakumar (2006), Novy Marx (2012)) shows that the abnormal returns to momentum portfolios disappear after controlling for measures of earnings surprises. Motivated by this evidence, this paper proposes a micro-founded model that generates earnings momentum and price momentum. The key feature of the model is that earnings are subject to both temporary and permanent shocks. I show that the combination of the two shocks matches the short-to-intermediate term nature of momentum profits. This paper examines the pricing implications of a firm s cash policy response to the two types earnings shocks, which, to the best of my knowledge, is an unexplored channel for the purposes of studying momentum. Since profits to momentum strategies are relatively short lived, the link between earnings and price momentum is more likely to emerge through the cash policy of a firm 1 Rouwenhorst (1998) shows that there is persistence in returns over the medium-term horizon not only in the US, but also in international equity markets. Moskowitz and Grinblatt (1999) show that there is a strong momentum effect in industry portfolios. Chan, Hameed and Tong (2000) show that there is momentum also in international stock market indices. Jagageesh and Titman (2001a) show that the profits to the momentum anomaly have not disappeared after its initial discovery (at the time the paper was published). Avramov et al. (2007) show that momentum profits arise mainly due to low rated firms, while there is no momentum present among high rated firms. Asness, Moskowitz and Pedersen (2013) document the presence of momentum across eight different markets and asset classes. 2 Data mining has also been proposed as a possible reason for the existence of momentum returns. However, the pervasive evidence on the profitability of momentum strategies in out-of-sample tests, both before and after the seminal Jegadeesh and Titman (1993) study, addresses any such concerns. 3 Johnson (2002) links momentum to the growth rate in dividends, which follows a mean-reverting process. He shows that sensitivity to dividend growth rate risk increases with growth rates, justifying the higher expected returns to the winner leg of the momentum portfolio. Berk, Green and Naik (1999) model betas as being dependent on the collection of projects the firm has invested in, a collection which changes slowly over time 2

3 rather than investment or capital structure policies which have a more long-term nature and do not adjust as frequently. Bolton, Chen and Wang (2011) show that cash policy is indeed important for firm valuation. Decamps et al. (2016) build a similar model but incorporate both temporary and permanent shocks to earnings and show that their combination has different implications on cash policy and valuation. The setting in this paper is thus similar to Decamps et al. (2016), with the addition of a stochastic discount factor that prices both temporary and permanent systematic shocks. Profitability-scaled cash holdings, which depends on both types of earnings shocks, drives the dynamics of the model. Exposure to transitory shocks can lead to losses, motivating the firm to maintain cash reserves. Holding cash is costly and there is an optimal or target level of cash that balances the precautionary benefits with the carry costs (Bolton, Chen and Wang, 2011). Whenever cash exceeds the optimal level, the firm distributes the excess as a dividend. When cash holdings fall below target, transitory shocks can potentially drive the firm out of business, regardless of its long term prospects. As a result, the firm becomes increasingly sensitive to transitory shocks. The sensitivity to permanent shocks, on the other hand, does not change significantly and becomes less important compared to the transitory one. Based on this analysis, the firms experiencing the largest price responses to cash-flow shocks are the ones where cash is below target. A sorting by past performance over the previous year will therefore contain a large proportion of this subset of companies. At the time of portfolio formation, the firms that have had the largest price increases (the winners) will have reached a level of productivity-scaled cash holdings that is close enough to target so that they are less likely to be liquidated. The losers on the other hand, are the surviving companies among the most constrained ones that have received negative earnings shocks. The model intuition is that momentum strategies are bets that the prices of winners will continue to rise due to them being closer to target, while the prices of the losers will continue to decline due to them being farther from target. Sorting on performance over the past year provides an indication as to how far from their respective targets these companies are. The closer to (farther from) the target, the more likely it is that price increases (declines) will continue. The first prediction of the model is therefore that momentum returns are attributable to those companies where scaled cash holdings are below target. A simple test in the data confirms this. A conditional double sort on momentum and a proxy for the distance from target cash shows that momentum portfolios yield statistically significant alphas only in the lowest quintile for the distance from target cash holdings. This result links momentum to financial distress, which is consistent with the finding in Avramov (2005) that momentum is concentrated in companies that have the lowest credit ratings. These represent a very small proportion of the universe of stocks, and so do companies whose scaled cash holdings are below target. Computing the correlation between realized cumulative excess returns and expected excess returns in model simulations allows for comparative statics exercises. These shed light on how firm fundamentals relate to the correlation in returns, therefore allowing to identify the companies 3

4 with the highest correlation. As argued by Sagi and Seasholes (2009), a higher autocorrelation in returns can provide enhanced momentum strategies since the winners (losers) with the highest autocorrelation will be more persistent. The model predicts that the correlation between realized and expected returns should be highest on those stocks that have: (1) the highest volatility of shortterm shocks, (2) the lowest volatilities of permanent shocks, (3) positively correlated permanent and short-term earnings shocks, (4) the lowest growth rates in productivity and (5) the lowest growth rates in productivity-scaled earnings. The higher the volatility of short-term shocks and the lower the volatility of productivity shocks, the more likely it is that the short-term beta of the company will be high. This provides a testable hypothesis in terms of momentum profits being higher for companies where the ratio of the volatilities of short-term and permanent shocks is higher. I use sales as a proxy for productivity, which serves to compute the volatility of permanent shocks. The volatility of the earnings proxies for the volatility of short-term shocks in the model. I then form portfolios by double sorting on past performance over the previous year and the ratio of the shock volatilities. The returns to momentum strategies along the quintiles with the highest volatility ratios produce higher returns compared to unrestricted momentum strategies. Returns to momentum portfolios formed on simulated model data with parameters set at the baseline levels close to those in Decamps et al. (2016), are positive and statistically significant. They average at 2% per month, with a test statistic greater than 4.5. In different sets of simulations, one of the key parameters of the model is changed in order to observe the effect on momentum portfolio returns. The results of these simulations are generally in line with those of the comparative statics exercises. Since the dynamics of the model are driven by earnings shocks, which in reality the firm receives on earnings announcement dates, the model links earnings announcements to price momentum. Constructing Standardized Unexpected Earnigns (SUE) based on the model, where earnings changes are linked to the level of productivity, yields a measure that, on its own, is able to reduce the power of momentum in cross-sectional Fama-MacBeth regressions. Accounting for profitability is therefore important for explaining the effect of past performance on expected returns. This provides additional support for the mechanism for momentum that is implied by this model. Another widely used measure of earnings surprises in the literature is Cumulative Abnormal Returns (CAR) on the earnings announcement day. According to the model, the price responses of liquidity constrained firms on the announcement day will be large due to the higher short-term beta. Therefore, these higher returns will not necessarily be abnormal, but a compensation for the higher liquidation risk. Independent double sorts on CAR and the proxy for the distance from target cash show that CAR strategy returns are higher in the quantile where firms are most constrained. This paper puts together various strands of the literature related to momentum, linking it simultaneously to time-variation in expected returns, earnings momentum and financial distress. Conrad and Kaul (1998) argue that differences in unconditionally expected returns yield momentum effects, since a sort on past performance would be a sort on these unconditional expected returns. 4

5 Jegadeesh and Titman (2001) criticise this conjecture stating that their model would imply that momentum profits would persist indefinitely, which is inconsistent with the evidence of the disappearing profitability of the strategy for holding periods beyond a year. Time-variation in expected returns makes the model here not subject to the same critique. Chordia and Shivakumar (2002) also link momentum to time-series variation in conditionally expected returns. These can be predicted by a set of macroeconomic variables, which are mainly related to credit market conditions. This is in line with Avramov (2005) linking momentum to financial distress. The paper is, however, mainly descriptive in the sense that it documents where momentum returns are highest but it does not explain why this occurs. The model presented here can provide a link between earnings momentum and price momentum that arises from financial distress, which could also explain the findings of Avramov (2005). Accounting for exposures to both permanent and short-term shocks is instrumental for the model being able to explain momentum strategy profits. Gorbenko and Strebulaev (2010) and Decamps et al. (2016) among others, stress the importance of both types of shocks in shaping firms financial policies. This paper is similar in spirit with Palazzo (2011), in terms of considering the effects of corporate cash holdings on equity risk premia based on a model where target cash holdings and risk depend on the correlation between cash flows and a pricing kernel. This paper is also related to Johnson (2002), where momentum effects arise from stochastic expected growth rates to dividends. Expected excess returns are also time-varying in this setting and, under certain assumptions, positively correlated with cumulative excess returns. Once generalized to account for both long-term and short-term shocks to the dividend growth rate, the model is much better capable of reproducing momentum effects, although of a much smaller magnitude than those observed in the data. Jegadeesh and Titman (1993) also recognize that there might be a link between news on short-term and long-term prospects of the company and the profitability of momentum strategies. They relate these to investor over/under-reaction. The analysis presented here provides an alternative explanation that is based on the company s time-varying exposures to systematic long-term and short-term risk. These two types of explanations need not necessarily be mutually exclusive. The rest of the paper is organised as follows. Section 2 presents the model setup and the beta pricing implications. Section 3 provides the results of the comparative statics exercises and model simulations, along with the testable predictions. Section 4 describes the data and identification procedure. Some preliminary results are presented in Section 5. Section 6 concludes. II. The Model A. Model setup The setup of the model is that of Décamps et al. (2016). Markets are complete and arbitragefree. Time is continuous and the risk-free rate is constant at r > 0. The firm considered in this model is an all equity-firm, whose cash-flows are exposed to both 5

6 permanent and transitory shocks. Shocks of a permanent nature affect the productivity of the firm s assets in place. This productivity is denoted by A t and is assumed to follow a geometric Brownian motion: da t = µa t dt + σ P A t dwt P where µ and σ P > 0 are constant and W P is a standard Brownian motion under the physical measure, P. The parameter µ represents the expected growth rate in the firm s productivity, while σ P is the volatility of the productivity process. The cash flow that is generated every period, denoted by dx t, is uncertain and depends on the level of productivity in the previous period: dx t = αa t dt + σ T A t dwt T where α and σ T are positive constants and W T is a standard Brownian motion under the physical measure, P. The parameter α represents the expected growth rate in productivity-scaled cash flows, while σ T is the volatility of the productivity-scaled cash-flow process. W T represents the short-term shock to scaled cash-flows, and it is correlated with W P with an instantaneous correlation coefficient of ρ [ 1, 1]: dw T t dw P t = ρdt Given this correlation, it is possible to decompose the short-term shocks to cash flows into permanent and transitory components: dwt T = ρdwt P + 1 ρ 2 dwt Z where W Z is another Brownian motion which is uncorrelated to W P. This means that shortterm shocks to cash flows (dwt T ) consist of a combination of shocks to the productivity level which have a permanent nature (given that productivity follows a GBM) and shocks of a transitory nature that do not necessarily affect productivity. The cash flow process can then be expressed as: dx t = αa t dt + σ T A t ρdw P t + σ T A t 1 ρ 2 dw Z t If the firm is not exposed to short-term shocks (σ T = 0), cash-flows cannot be negative. This is because in this case they would be given by αa t dt and both α and A t are positive. The presence of short-term shocks (σ T > 0) means that cash-flows can become negative. The firm is therefore exposed to potential losses, and has a precautionary motive for retaining earnings as cash reserves. The firm s cash holdings are denoted as M t, and there is a carry cost of liquidity denoted as λ where λ (0, r]. Cash reserves have the following P-dynamics: dm t = (r λ)m t dt + dx t dd t 6

7 where D t is the cumulative dividend paid to shareholders up to time t. The firm is liquidated at time τ if the cash buffer reaches zero following a series of negative shocks. The firm value will then be a function of productivity and cash reserves, V (a, m), and it will be given by: [ τ ( )] ω V (a, m) = max (D EQ a,m e rt dd t + e rτ ˆαAτ t) t,τ 0 r ˆµ + M τ where ω is the fraction of the unconstrained value of the assets that is recovered in the liquidation event, ˆα and ˆµ are the risk-adjusted growth rated in cash flows and productivity respectively. The objective of the shareholders is to choose the dividend and liquidation policies that maximize firm value. B. Model solution In the region where it is optimal to retain earnings, M (0, M), the equity value function V (a, m) will satisfy the following ODE: rv = ˆµaV a + (ˆαa + (r λ)m)v m a2 ( σ 2 P V aa + 2ρσ P σ T V a,m + σ 2 T V mm ) The LHS of the above equation represents the required return on the equity of the firm. The first two terms on the RHS represent the effects of changes in profitability µa and cash savings αa + (r λ)m. The last term represents the effects of the volatilities in profitability and cash flows. V a,m 0 in this model, meaning that changes in productivity affect firm value as well as cash reserves. The equity value is homogenous of degree one in A and M, therefore: V (a, m) = av (1, m ) af (c) a where c = m a represents the productivity scaled cash holdings. The first and second order derivatives of the equity value with respect to productivity and cash holdings can be expressed as: V a = F (c) cf (c), V aa = c2 a F (c), V m = F (c), V mm = 1 a F (c) and V am = c a F (c). The above ODE can then be re-written as: (r µ)f (c) = (ˆα + (r λ ˆµ)c)F (c) subject to boundary conditions F (0) = ω ˆα r ˆµ, ( σ 2 P c 2 2ρσ P σ T c + σt 2 ) F (c) (1) F (c ) = 1, F (c ) = 0, F (c) = F (c ) + c c, for c > c 7

8 . C. Expected returns and risk premia In order to analyse expected returns and risk premia under this setting, a representative agent is assumed to have a marginal utility process Λ t, whose dynamics are given by: dλ t Λ t = rdt η T dz T t η P dz P t where Z T t and Z P t are standard Brownian motions independent of one another. Z T t is correlated with the source of short-term risk to the firm s cash-flows, W T t, with a correlation coefficient of χ T. Z P t is correlated with the source of permanent risk to the firm s cash-flows, W P t, with a correlation coefficient of χ P. η T and η P are the market prices of short-term and permanent cash-flow risks, respectively. This specification of the stochastic discount factor implies that the systematic components of both short-term and permanent sources of risks are priced. In order to derive the conditional risk premium on the equity, coefficients in the ODE in Equation (??) can be compared to the coefficients of the HJB equation for F (c) under the physical measure. It can be shown that the conditional expected excess return on the equity, denoted as EER t, is given by: EER t (c) = χ T σ T η T F (c) F (c) + χ P σ P η P (1 cf (c) F (c) ) (2) The above expression shows that the equity s conditional risk premium is given by the sum of the risk premiums associated with exposures to permanent and short-term systematic risks. The short-term shock premium is given by the first term on the right-hand side of Equation (??). It is determined by the market price of short-term cash-flow risk, η T, and the firm s exposure to this risk. The latter is given by the product of the correlation of the firm s cash-flows to systematic shortterm cash-flow shocks, χ T, the volatility of the firm s scaled cash-flows, σ T, and the semi-elasticity of F (c) with respect to c, F (c) F (c). The permanent shock premium is given by the second term in Equation (??). It is determined by the market price of permanent risk, η P, and the firm s exposure to this risk. The exposure to permanent shock risk is given by the product of the correlation of the firm s cash-flows with permanent shocks to the pricing kernel, χ P, the volatility of the productivity process, σ P, and 1 minus the semi-elasticity of F (c) with respect to c, 1 cf (c) F (c). This specification implies that conditional expected returns are time-varying and depend on the level of productivity-scaled cash holdings. D. Expected returns and cumulative realized returns For the purposes of studying momentum, it is interesting to see the conditions under which the covariance between expected excess returns (EER t ) and cumulative excess returns (denoted as 8

9 CER t ) is positive. The instantaneous cumulative excess return will be given by: dcer t (c) dv V rdt = {ˆµ(1 cf (c) F (c) ) + (ˆα + (r λ)c)f (c) F (c) + 1 [ σ 2 2 P c 2 2ρσ P σ T c + σt 2 ] F (c) F (c) r}dt + (1 cf (c) F (c) )σ P dw P t + F (c) F (c) σ T dw T t (3) The instantaneous covariance between realized and expected returns is given by: E t [(CER t+dt E t (CER t+dt )) (EER t+dt E t (EER t+dt ))] (4) The overall sign of the covariance will depend on the signs of the correlations between transitory and permanent cash flow shocks with transitory and permanent shocks to the pricing kernel. Looking at the instantaneous correlation coefficient between cumulative and expected returns (denoted as Υ(c)) is, however, more informative as many terms simplify and it becomes easier to determine conditions under which the expression would be expected to be positive. Denoting f 1 (c) = F (c) F (c), it can be shown that the instantaneous correlation between cumulative and expected returns is given by: Υ(c) = 1 f 1 (c)σc 2 C t σp 2 + σ P σ T ρ σ c [f 1 (c)] 2 σt 2 + (1 c f 1(c)) 2 σp 2 + 2f 1(c)(1 c f 1 (c))σ P σ T ρ (5) The sign of Υ(c) is determined by the sign of the numerator in (5). Depending on the correlation coefficient between permanent and transitory shocks: For ρ > 0, Υ(c) > 0 if f 1 (c) > 1 σ 2 c ( Ct σ 2 P σ P σ T ρ ). For ρ < 0, Υ(c) > 0 if f 1 (c) > 1 σ 2 c ( Ct σ 2 P + σ P σ T ρ ). In both cases, the instantaneous correlation between cumulative past and expected returns would be positive when the term f 1 (c) is sufficiently large. This would imply that positive correlation in returns would be expected when f 1 (c) is large. As will be shown in the comparative statics exercises in the next section, the correlation between cumulative excess returns and expected excess returns is higher for firms where the correlation between permanent and transitory shocks is positive. In this case, the effects of permanent shocks will be amplified by the positively correlated transitory shocks (when the firm is constrained) and this will be more so for firms where their short-term beta is higher. 9

10 E. Momentum mechanism A beta pricing model can be derived from this setting assuming that there is a traded asset (such as the market) whose returns follow a Brownian motion with a drift. It can be shown that the short-term beta can be expressed as: β T t (c) = χ T σ T σ T M The permanent-shocks beta can be expressed as: β P t (c) = χ P σ P σ P M F (c) F (c) ( 1 cf ) (c) F (c) Both betas vary over time and depend on the level of productivity-scaled cash holdings. leads to time-variation in the short-term shock beta. The ratio is positive and decreasing (proof provided in Appendix??). As a result, the transitory beta rises with the negative of the distance of scaled cash holdings from the target level. cf (c) F (c) leads to time-variation in the permanent shocks beta. The sign of cf (c) F (c) and the sign of its derivative are, on the other hand, unconstrained (proofs provided in Appendix??). F (c) F (c) also represents the semi-elasticity of firm value with respect to scaled cash holdings in the region (o, c ). Similarly to the transitory beta, the sensitivity of firm value to cash rises with the distance of cash holdings to target. This sensitivity is greater than exponential. As argued by Johnson (2002), extreme sensitivity of firm value with respect to a risk factor may cause prices to behave in a fashion that seems bubble-like but is in fact rational. Figure?? illustrates the behaviour of the transitory and permanent betas. The left panel shows how the permanent and transitory betas change with scaled cash holdings (when below target) for a firm that approaches liquidation. The right panel shows the corresponding change in expected excess returns and the instantaneous correlation between cumulative and expected returns. The parameters are set at the baseline levels of Decamps et al. (2016). The volatilities of short-term and permanent shocks of the market are set to: σm T = 0.09 and σt P = 0.25 and the correlations of the firm s cash flows to short-term and permanent shocks to the pricing kernel are both set to be equal to 0.8. This is so as to be able to compare the two betas, βt T and βt P, only along their respective sensitivities to productivity scaled cash holdings. The number of months used in the simulation is 600. F (c) F (c) 10

11 T (c) t P (c) t 0.7 EER t (c) (c) Beta 0.6 EER t (c) Scaled cash holdings Scaled cash holdings Figure 1. The figure plots the permanent and transitory betas as a function of productivity-scaled cashholdings. The parameter values are at the baseline level of Decamps (2016), where:α = 0.18, µ = 0.01,σ P = 0.25, σ T = 0.09, ρ = 0.5, r = 0.06 and λ = The market parameter values are set to: σ T M = 0.09 and σ T P = The correlations between long-term and short-term shocks to the pricing kernel and long-term and short-term shocks to cash flows, denoted as χ P and χ T respectively, are both set to 0.8 in both cases. The left panel in Figure?? shows that the permanent shock beta is more important at higher levels of scaled cash holdings. The short-term shock beta, on the other hand, increases at an increasing rate with the distance from target and becomes much larger than the permanent shocks beta. When cash reaches very low levels, transitory shocks have a larger effect on scaled cash holdings. In this case, the cash balance is low relative to productivity. A transitory shock (either positive or negative) affects only the numerator of the scaled cash holdings ratio, while a permanent shock affects both numerator and denominator. At a low level of the numerator, transitory shocks affect the ratio of cash to profitability more. If the cash balance was already high, the effects of transitory shocks would be much smaller. Intuitively, firms that are close to liquidation are extremely sensitive to transitory cash-flow shocks as these could potentially lead to immediate liquidation, regardless of the long-term prospects of the company. The right panel in Figure?? shows expected returns rising with the (negative of ) the distance of profitability scaled cash holdings from their target level. Similarly, the correlation between cumulative excess returns and expected excess returns (plotted on the right vertical axis) increases with the distance from target cash. The correlation is positive in all instances, implying that high (low) expected excess returns follow high (low) cumulative realized excess returns for this firm. In the region where the short-term beta exceeds the permanent one, the correlation plot steepens. The result in Equation (??) highlights the need for a sufficiently large transitory beta for a positive instantaneous correlation between cumulative and expected returns. The plots in Figure?? 11

12 generally conform with the implication of Equation (??). The figure also shows that when cash holdings approach the target, the correlation between expected and past returns reaches a low level. Because empirically most firms maintain cash holdings at the target level, the model simulations imply a low correlation coefficient for most firms. In other words, a significant positive correlation between cumulative and expected returns would be expected only for the most constrained firms. As mentioned earlier, the normalized transitory shock beta also represents the semi-elasticity of firm value with respect to scaled cash holdings. Both are convex. The semi-elasticity of firm value with respect to cash represents the return as a response to changes in cash. The convexity of this semi-elasticity, essentially representing the sensitivity of returns to changes in cash, implies increasingly larger returns in absolute terms as the firm becomes more constrained. The largest recent price moves (highest returns in absolute terms) will therefore occur in those firms where the distance of their scaled cash holdings from their target has been highest. Based on the above analysis, momentum would be expected to be concentrated on the most constrained, high short-term beta firms. Among these, because the firm value function F(c) is increasing in c, the firms with positive cumulative returns (i.e. the winners) are the ones where scaled cash holdings have increased. Avramov (2002) documents positive sales growth for the winners in the year prior to portfolio formation. Interpreting sales as a profitability indicator, large positive permanent (profitability) shocks most likely drive the increase in the scaled cash holdings of the winners. The effects of permanent shocks can be analysed looking at the dynamics of scaled cash holdings: dc t = [α σ P σ T ρ + (r λ µ)c t ]dt + σ T 1 ρ 2 dw Z t + (ρσ T C t σ P )dw P t dd t A t (6) A positive permanent shock leads to an increase in scaled cash holdings when the correlation between permanent and short term shocks is positive and scaled cash holdings are at a low level (which is the case in the analysis here). Hence the winner firms are more likely to have a positive correlation between permanent and transitory shocks. At the time of portfolio formation, a momentum sort will go long the stocks that have had the largest price increases, which are likely to be the most constrained firms that have had the largest increases in scaled cash holdings. These will most likely be firms that have positively correlated short-term and permanent shocks. Intuitively, the largest increase in scaled cash holdings will occur when these positive realizations of permanent shocks have coincided with positive transitory shocks. At the time of portfolio formation, since all shocks are IID, the firm is equally likely to receive either positive or negative shocks of a transitory or permanent nature. Considering the fact that the winners in the past year are constrained firms that have received the largest positive shocks of both types, their level of productivity is most likely to have increased to a high enough level where there isn t much convexity in the short-term beta. From Equation (??), when C t is high enough, the change in scaled cash holdings for these firms is 12

13 more likely to be positive than negative. Scaled cash holdings will most likely increase regardless of the sign of the shock of either type that the firm receives. Some of the winner firms, however, may not have reached this level of productivity. In this case, the price decrease from a decline in scaled cash holdings will be larger in absolute terms than the price increase from an increase in scaled cash holdings (due to the convexity). A decrease in scaled cash holdings will increase the transitory beta and correspondingly the expected return on the stock (instantaneous or for a given holding period). On average, most winners will be expected to have reached a high enough productivity level whereby it is more likely that scaled cash holdings increase and hence firm value increases. This occurs in the region where the short-term beta function becomes flat. Because some of the past winners may not have reached such a level, however, and experience reversal as a result, the overall expected return on the winner portfolio increases. Considering the past winners that reversed at time t, half of them will have a decline in price while the other half an increase. As a result, the prices of most winners in the portfolio will continue to increase over the holding period. Intuitively, since the past winners are most likely constrained firms that have received positive permanent shocks, the winner leg of a momentum portfolio would be a bet that the prospects of these firms, on average, will continue to improve. On the other hand, firms with negative cumulative returns (i.e. the losers) would most likely be those that have recently received large negative productivity shocks and have a positive correlation between permanent and transitory shocks. This is consistent with Avramov (2005) documenting negative sales growth for the loser firms in the year prior to portfolio formation. If the correlation were negative, because of the low level of scaled cash holdings, permanent and transitory shocks of opposite signs would offset each other and thus scaled cash holdings would not change much. A positive correlation makes it likely that negative productivity shocks are associated with negative transitory shocks, leading to a larger decrease in scaled cash holdings. At the time of portfolio formation the level of scaled cash holdings of the losers will be quite low. Due to the very high convexity at such low levels of scaled cash holdings, decreases in the cash ratio will lead to larger price declines in absolute terms than the price increases from a rise in scaled cash holdings. Large enough shocks at this point are very likely to lead to liquidation. In this setting, due to the convexity of the beta (and the firm s semi-elasticity with respect to scaled cash holdings) firms that are liquidated in any given period are the past losers. At the time of portfolio formation, the biggest losers of the past year will have gone out of business and the survivors will be the ones that have had a recent increase in scaled cash holdings (short-term reversal). This means that the expected return on the companies that have survived up to time t will be lower. Due to the convexity in the beta, however, on average the price of the portfolio of losers will continue to decline. In this case there is also a positive relationship between realized cumulative returns over the previous year (skipping the most recent month) and expected excess return on the company. Going short these companies then is essentially a bet that despite the recent positive liquidity shock, the firm is not a good investment over the long term. The main takeaway from the above analysis is the concentration of momentum sorts on liquidity 13

14 constrained firms. The model proposes a mechanism for the emergence of momentum that is based on distress risk, in accordance with Fama and French (1992) relating cross-sectional anomalies in returns to financial distress. The analysis also supports the finding in Avramov (2005) that momentum is concentrated among firms with the highest credit risk. The model does not incorporate debt, but a large corporate finance literature considers cash holdings as negative debt. A decline in the level of cash, therefore, would be equivalent to an increase in the level of debt (from target). The most constrained firms would be the riskiest ones, and would therefore be the ones with the lowest credit ratings. This is consistent with Acharaya and Davydenko (2012) who show that, over the short term, there is a negative correlation between cash holdings and credit risk. Given the empirical evidence that most firms are close to their target level of cash holdings, a small proportion of firms would therefore be at levels very much below target. This is also consistent with Avramov (2005), who finds that it is only a small number of stocks that accounts for most of the momentum effects. III. Simulations A. Comparative statics Because the correlation between past and expected returns depends on the parameters governing the cash flow process, which also affect the state variable in the shareholders optimization problem, comparative statics would be instructive. Namely, it is useful to examine how changing some of the key model parameters affects the correlation in returns. These exercises allow for the identification of those firms where the correlation in returns is highest. As argued by Sagi and Seasholes (2007), restricting a momentum strategy to these firms would yield even higher momentum returns than those identified by Jegadeesh and Titman (1993). The returns to such strategies would be higher because there would be more persistence in both winners and losers. The baseline parametrization in Décamps et al. (2016) serves as the baseline in the simulations presented here as well. In each of the comparative statics exercises presented in Figure?? one of the key parameters varies over a range of plausible values presented on the horizontal axis (expressed in annual terms). The choices of the supports for the parameter values generally rely on the estimation results from Gryglewicz et al. (2017). In each simulation, solving the model at every point in time allows for the computation of the correlation between past and expected returns (at every point in time). So as to control for the different paths that the productivity-scaled cash holdings can take, the simulation is repeated 100 times for a given parameter set. The plots report the average over each of the 100 simulations of the time-series averages of the correlation in returns. The first panel in Figure?? shows how the instantaneous correlation between expected and cumulative past returns changes with the correlation between permanent and short-term shocks. The black line represents the baseline case, where the correlation between cash-flow shocks is set to 0.2. The average correlation coefficient between expected and past returns increases with the 14

15 = -0.2 T = 0.25 P = 0.5 = Correlation between productivity and cash flow shocks, = -0.2 = 0 =0.5 = Volatility of short-term shocks to cash flows, T = -0.2 = 0 =0.5 = Volatility of long-term shocks to cash flows, P = = 0 =0.5 = Expected productivity growth rate, = -0.2 = 0 =0.5 = Expected cash flow growth rate, Figure 2. Comparative statics: Changes in correlation between cumulative past and expected returns with respect to model parameters. The figure shows the average instantaneous correlation between cumulative excess returns and expected excess returns using simulated data. The correlation coefficient is plotted as a function of the volatilities of long-term and short-term cash flow shocks, the expected cash flow and productivity growth rates and the correlation coefficient between temporary and permanent shocks. The baseline parametrisation in each plot is similar to Décamps et al. (2016), where r = 0.06, α = 0.18, µ = 0.01, σ P = 0.25, σ T = 0.18, ρ = 0.2, λ = 0.02 and ω = The correlations between long-term and short-term shocks to the pricing kernel and long-term and short-term shocks to cash flows, denoted as χ P and χ T respectively, are both set to 0.5 in all the simulations. 15

16 correlation between temporary and permanent shocks to cash flows. In expectation, returns reverse when the correlation between the cash flow shocks is negative and continue when the correlation is positive (Appendix?? provides analytical expressions for the correlation in returns in corner cases and when the correlation between the shocks is zero). Uncorrelated cash flow shocks are associated with an average correlation in returns close to zero. Gryglewicz et al. (2017) estimate the correlation coefficient between permanent and transitory shocks at an average value of One would, therefore, expect a low correlation between cumulative and past returns for the average firm. As argued in the previous section, a positive correlation between permanent and short-term shocks makes it more likely that the largest winners (losers) continue to win (lose) due to the dynamics of scaled cash holdings shown in Equation (??). Positively correlated shocks ensure that, at a high enough level of scaled cash holdings, price increases continue and that, at a low enough level of scaled cash holdings, price decreases continue. The dotted black line in the first panel corresponds to the case where the volatility of shortterm shocks is at a higher level (0.25) compared to its value in the baseline case. A relatively small increase in the short-term shock volatility leads to an upward parallel shift of the line. For any given level of correlation between cash-flow shocks, a higher volatility of short-term shocks makes it more likely that the correlation in returns is positive. When increasing the other parameters, namely σ P, α and µ, the line shifts downwards. This would imply that the volatility of permanent shocks and the growth rates of the productivity and scaled cash-flow processes have a negative effect on the correlation in returns for any given level of correlation between cash-flow shocks. The effects of each of these parameters are shown separately in the other four panels in Figure??. The second panel in Figure 1 shows how the average correlation in past and expected returns changes with respect to the volatilities of short-term shocks, while the third panel shows how the average correlation in past and expected returns changes with respect to the volatilities of permanent shocks. The effects of the two are opposite. The higher the volatility of short-term cash-flow shocks, the higher the instantaneous correlation between cumulative and expected returns. Intuitively, firms with a higher volatility of short-term shocks would have a higher short-term beta, all else equal. This would make them more susceptible to transitory shocks in case their scaled cash holdings fall below target. The short-term beta of these firms would then become even larger. As argued in the previous section, there is a positive relationship between the correlation in returns and the short-term beta of the firm. The plot in the third panel shows a different picture. Expected returns follow cumulative returns less closely for firms with more volatile productivity shocks. As argued above, winners and losers in a momentum sort most likely have positively correlated cash-flow shocks. For firms with positively correlated cash flow shocks, target scaled cash holdings decrease with the volatility of permanent shocks. In this case, a low volatility of permanent shocks increases the target. Because a higher target denotes a riskier firm, the riskiest firms in a momentum sort have low volatilities of permanent shocks. Since the permanent shock volatility is low, the permanent shock beta is likely 16

17 to be low. As a result, the short-term shock beta will most likely have a larger weight in the firm s overall beta. As argued above, a sufficiently high short-term beta allows for positively correlated cumulative past and expected returns. This correlation in returns is increasingly positive the larger the short-term beta becomes with respect to the permanent one. The red line in the same plot presents the case corresponding to uncorrelated cash-flow shocks. This line is above the one representing the baseline case (where ρ = 0.2), consistent with the predictions in the first panel of the figure. The dotted black line represents the case where the correlation in the cash-flow shocks is even higher (set at ρ = 0.5), and correspondingly the line shifts upward. The dashed black line represents the case where the correlation in -0.5 and as a result the correlation between cumulative past and expected returns is positive only when the volatility of permanent shocks is lowest. Appendix?? shows the analytical expressions for the correlation in returns in the extreme cases for the volatilities of permanent and short-term shocks. The fourth panel in Figure?? shows how the correlation in expected and cumulative excess returns changes depending on the expected growth rate in productivity. Returns persist less at higher levels of the productivity growth rate. A similar relationship holds between the expected growth rate in cash flows (fifth panel in Figure??) and the correlation in returns. Firms with the highest growth rates in productivity and earnings would be perceived as less risky, implying lower expected returns. This means that, in both cases, low expected excess returns follow high cumulative returns. These firms are also more likely to be close to their respective target levels of scaled cash holdings, where the correlation in returns is generally low. B. Model simulation Table 1 shows the performance of momentum strategies constructed on simulated panel data. For each parameter set, I simulate 100 panels and report the average returns and average test statistics of the 100 momentum portfolios. The chosen number of simulations for each parameter set ensures that the average value of the momentum mean portfolio returns falls within a 95% confidence interval. Each panel consists of the returns of 2000 firms simulated over 600 months, dropping the first 200 months to ensure that a steady state distribution is reached. Fama and French (1992) use a dataset of similar size for their empirical investigations. Simulating a panel dataset requires firms differing in some characteristic. I draw the correlation coefficients of firms cash-flows to permanent and short-term shocks to the pricing kernel from beta distributions. For each firm, the correlation between permanent cash-flow shocks and permanent shocks to the pricing kernel and the correlation between transitory cash-flow shocks and transitory shocks to the pricing kernel are drawn independently. 90% of each type of correlation (permanent or transitory) are drawn from a beta distribution with shape parameters: α χ = 3 and β χ = 5. 10% of each type of correlation (permanent or transitory) are drawn from the negative of a beta distribution with shape parameters: α χ = 1 and β χ = 3. For each type of correlation, the resulting distribution of all the observations, although defined over the interval [-1,1], resembles a normal. 17

18 Table I. This table reports the average mean returns and average test statistics of momentum portfolios constructed on 100 simulated panels for each scenario. Scenario 3 draws the volatilities of transitory cashflow shocks from a uniform distribution with support [0.05, 0.25]. The fourth scenario draws the volatilities of the permanent cash-flow shocks from an exponential distribution where the rate parameter is given by the inverse of the difference between an assumed mean of 1 (close to the estimate from Gryglewiz et al., 2017) and a lower limit of 0.1. Scenario 5 draws the volatilities of the permanent cash-flow shocks from a uniform distribution with support [0.01, 2]. Scenario σ T σ P ρ µ α m t-stat Winner return Loser return U(0.05, 0.25) exp(1.11) U(0.01,2) The results reported in Table?? are in line with the intuition gained from the comparative statics exercises. The first and second scenarios yield statistically significant momentum returns on average. Profits to momentum strategies average close to 2% per month with a t-stat > 4.5 in both cases. A higher volatility of short-term shocks in the second scenario yields higher average returns and higher test statistics on average than the first scenario. Although still significant, momentum strategies in the third scenario have lower average returns. Two main reasons lead to lower momentum returns. First, drawing the transitory cash-flow shock volatility from a uniform distribution with support [0.05, 0.25] leads to the ratio of the transitory shock volatility to permanent shock volatility being lower for most firms. The lower volatility ratio diminishes the importance of the transitory shock beta. In this case, the convexity necessary to generate momentum effects has a lesser impact. The second reason for the lower returns relates to the negative correlation between transitory and permanent cash-flow shocks. As argued in the section on the mechanism for momentum, a positive correlation between the shocks ensures continuation of returns for winners and losers. The negative correlation affects the loser leg of the portfolio more. The loser leg drives most of the momentum returns in the first two scenarios while in the third the returns for the losers turn from negative to positive. The fourth and fifth scenarios incorporate a higher permanent shock volatility, resulting in insignificant average momentum strategy returns. A higher permanent shock volatility also lowers the relative importance of the transitory shock beta. The last scenario examines the effects of the expected growth rates in productivity and cash-flows. When set at higher levels, momentum returns diminish further. Although some of the scenarios produce significant momentum effects, I do not make any claims regarding these simulations truly being representative of what happens in reality as it is generally quite difficult to model the full covariance structure of returns. Namely, the assumptions on the distributions of the correlation coefficients of firm cash flows to the pricing kernel are rather strong. They are based on evidence of CAPM-betas having a similar distribution, which could, on the other 18